Question: Solve for $X$. $X-\left[\begin{array}{rr}8 & -5 & 4 \\ 3 & -9 & 2 \\0 &1 &-2\end{array}\right]=\left[\begin{array}{rr}4 & 12 & -12 \\ 7 & 0 & -1 \\8 &2 &2\end{array}\right] $ $X=$
Explanation: The Strategy First, we can represent the matrices of the equation with letters, which will make the equation easier to handle. Then we can solve the equation for $X$ and obtain an expression with the letters we defined. Finally, we can substitute back the actual matrices into the resulting expression and simplify it. Solving the equation for $X$ We are given the following equation. $X-\left[\begin{array}{rr}8 & -5 & 4 \\ 3 & -9 & 2 \\0 &1 &-2\end{array}\right]=\left[\begin{array}{rr}4 & 12 & -12 \\ 7 & 0 & -1 \\8 &2 &2\end{array}\right] $ Let's represent the above matrices as follows. $A=\left[\begin{array}{rr}8 & -5 & 4 \\ 3 & -9 & 2 \\0 &1 &-2\end{array}\right]~~~~~~~~~ B =\left[\begin{array}{rr}4 & 12 & -12 \\ 7 & 0 & -1 \\8 &2 &2\end{array}\right] $ Then we can rewrite the equation as follows. $X-A=B$ Now it's simple to solve the equation for $X$. $\begin{aligned}X-A&=B\\\\ X&=A+B \end{aligned}$ Finding $X$ We found that $X=A+B$. Now we can substitute the actual matrices back into the expression and simplify. $\begin{aligned}X&=A+B \\\\&=\left[\begin{array}{rr}8 & -5 & 4 \\ 3 & -9 & 2 \\0 &1 &-2\end{array}\right] +\left[\begin{array}{rr}4 & 12 & -12 \\ 7 & 0 & -1 \\8 &2 &2\end{array}\right] \\\\\\&=\left[\begin{array}{rr}(8+4) & (-5+12) & (4-12) \\ (3+7) & (-9+0) & (2-1) \\(0+8) &(1+2) &(-2+2)\end{array}\right] \\\\\\&=\left[\begin{array}{rr}12 & 7 & -8 \\ 10 & -9 & 1 \\8 & 3 &0\end{array}\right]\end{aligned}$ Summary $X=\left[\begin{array}{rr}12 & 7 & -8 \\ 10 & -9 & 1 \\8 & 3 &0\end{array}\right]$